The utilization definition can also be related to Vapnik-Chervonenkis (VC) dimension of a class of binary classifiers \(H\)  from the image space \(\chi \)  to the label space \(\lbrace 0,1\rbrace \) ; \(H: \chi \rightarrow \lbrace 0,1\rbrace \)  [1]}, [2]}. The VC dimension \(VCdim(H)\)  is defined as the size \(m\)  of the largest shattered set of inputs \(\lbrace \vec{x}_{1},..., \vec{x}_{m}\rbrace \)  such that the growth function
\(\Pi _{H}(m)= \max _{\vec{x}_{1},..., \vec{x}_{m} \in \chi } | \lbrace  (h(\vec{x}_{1}),..., h(\vec{x}_{m}) ): h \in H \rbrace  |= 2^{m}\)  [2]}.
For real-valued functions \(\mathcal {F}\)  present in neural networks, one can define \(VCdim(\mathcal {F}) = VCdim(\lbrace sgn(f):f \in \mathcal {F}\rbrace \) , which is achieved by thresholding at zero in the utilization computation. The pseudodimension of \(\mathcal {F}\)  or \(Pdim(\mathcal {F})\)  has been proven to satisfy \(VCdim(\mathcal {F}) \le Pdim(\mathcal {F})\)  [4]}.
For example, a convolutional computation unit \(v_{j}\)  with the output tensor \((Channels, Width, Height)\)  per input image \(\vec{x_{i}}\)  would yield \(VCdim(\mathcal {F}(v_{j}))=Channels\)  after binarizing the output values. The \(VCdim\)  values are used to quantify utilization of each computation unit as documented in Section . Our work does not currently leverage the possibility of using lower and upper bounds on \(VCdim(\mathcal {F})\)  for an entire neural network. These bounds have been derived for AI architectures with computation units characterized by constant and known non-linearities (piece-wise constant, piece-wise linear, and piece-wise polynomial [2]}) with respect to the number of layers \(L\)  and parameters \(W\) . While in practice there is no guarantee that computation units in one architecture are not characterized by a mixture of non-linearity types, it would be possible to verify the theoretical bounds for a combination of \(L\)  and \(W\)  in a variety of existing popular AI architectures.
